Peter Jipsen Chapman University, Orange
Chapman University, Orange
Algebra, Logic and Foundations of Mathematics, Theory of Computation
Ph. D.
Publications
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ABSTRACT: We introduce concurrent Kleene algebra with tests (CKAT) as a combination of Kleene algebra with tests (KAT) of Kozen and Smith with concurrent Kleene algebras (CKA), introduced by Hoare, Möller, Struth and Wehrman. CKAT provides a relatively simple algebraic model for reasoning about semantics of concurrent programs. We generalize guarded strings to \emph{guarded seriesparallel strings}, or gspstrings, to give a concrete language model for CKAT. Combining nondeterministic guarded automata of Kozen with branching automata of Lodaya and Weil one obtains a model for processing gspstrings in parallel. To ensure that the model satisfies the weak exchange law $(xy)(zw)\leq(xz)(yw)$ of CKA, we make use of the subsumption order of Gischer on the gspstrings. We also define \emph{deterministic} branching automata and investigate their relation to (nondeterministic) branching automata. To express basic concurrent algorithms, we define concurrent deterministic flowchart schemas and relate them to branching automata and to concurrent Kleene algebras with tests.  [Show abstract] [Hide abstract]
ABSTRACT: Lattices have many applications in mathematics and logic, in which they occur together with additional operations. For example, in applications of Hilbert spaces, one is often con cerned with the lattice of closed subspaces of a fixed space. This lattice is not distributive, but there is an operation taking a given subspace to its orthogonal subspace. More gen erally, ortholattices are lattices with a unary operation ( )y that is involutive (a = ayy), sends finite joins to meets and for which a and ay are complements. Bounded modal lat tices (L;_;^; 0; 1; ;2) are models of (not necessarily distributive) modal logic, where and 2 are unary operations that preserve finite join and finite meet, respectively, and represent possible and necessary. Bounded latticeordered monoids are bounded lattices with an associative binary operation and an identity element 1. In these examples it is postulated that the additional operations "preserve structure" in various different senses. Orthocomplementation sends finite joins to meets (and finite meets to joins). The modal operators preserve finite joins and finite meets, respectively. Similarly, the monoid oper ation distribute over finite joins. Bounded residuated lattices are bounded latticeordered monoids with two further operationsn, = that interact with via the universally quantified residuation law: 
Article: Concurrent Kleene Algebra with Tests
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ABSTRACT: Concurrent Kleene algebras were introduced by Hoare, Möller, Struth and Wehrman in [HMSW09, HMSW09a, HMSW11] as idempotent bisemirings that satisfy a concurrency inequation and have a Kleenestar for both sequential and concurrent composition. Kleene algebra with tests (KAT) were defined earlier by Kozen and Smith [KS97]. Concurrent Kleene algebras with tests (CKAT) combine these concepts and give a relatively simple algebraic model for reasoning about operational semantics of concurrent programs. We generalize guarded strings to guarded seriesparallel strings, or gspstrings, to provide a concrete language model for CKAT. Combining nondeterministic guarded automata [Koz03] with branching automata of Lodaya and Weil [LW00] one obtains a model for processing gspstrings in parallel, and hence an operational interpretation for CKAT. For gspstrings that are simply guarded strings, the model works like an ordinary nondeterministic guarded automaton. If the test algebra is assumed to be {0,1} the language model reduces to the regular sets of boundedwidth spstrings of Lodaya and Weil. Since the concurrent composition operator distributes over join, it can also be added to relation algebras with transitive closure to obtain the variety CRAT. We provide semantics for these algebras in the form of coalgebraic arrow frames expanded with concurrency. 
Conference Paper: 14th International Conference, RAMiCS 2014
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ABSTRACT: Modular lattices, introduced by R. Dedekind, are an important subvariety of lattices that includes all distributive lattices. Heitzig and Reinhold developed an algorithm to enumerate, up to isomorphism, all finite lattices up to size 18. Here we adapt and improve this algorithm to construct and count modular lattices up to size 23, semimodular lattices up to size 22, and lattices of size 19. We also show that $2^{n3}$ is a lower bound for the number of nonisomorphic modular lattices of size $n$.  [Show abstract] [Hide abstract]
ABSTRACT: Residuated frames provide relational semantics for substructural logics and are a natural generalization of Kripke frames in intuitionistic and modal logic, and of phase spaces in linear logic. We explore the connection between Gentzen systems and residuated frames and illustrate how,frames provide a uniform treatment for semantic proofs of cutelimination, the finite model property and the finite embeddability property. We use our results to prove the decidability of the equational and/or universal theory of several vari eties of residuated latticeordered groupoids, including the variety of involutive FLalgebras. Substructural logics and their algebraic formulation as varieties of residuated  [Show abstract] [Hide abstract]
ABSTRACT: This paper studies generalizations of relation algebras to residuated lattices with a unary De Morgan operation. Several new examples of such algebras are presented, and it is shown that many basic results on relation algebras hold in this wider setting. The variety qRA of quasi relation algebras is defined and shown to be a conservative expansion of involutive FLalgebras. Our main result is that equations in qRA and several of its subvarieties can be decided by a Gentzen system, and that these varieties are generated by their finite members. 
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ABSTRACT: It is proved that any latticeordered pregroup that satisfies an identity of the form x ll···l = x rr···r (for the same number of l,r operations on each side) has a lattice reduct that is distributive. It follows that every such ℓpregroup is embedded in an ℓpregroup of residuated and dually residuated maps on a chain. 
Conference Paper: Categories of Algebraic Contexts Equivalent to Idempotent Semirings and Domain Semirings
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ABSTRACT: A categorical equivalence between algebraic contexts with relational morphisms and joinsemilattices with homomorphisms is presented and extended to idempotent semirings and domain semirings. These contexts are the Kripke structures for idempotent semirings and allow more efficient computations on finite models because they can be logarithmically smaller than the original semiring. Some examples and constructions such as matrix semirings are also considered. 
Article: Preface

Chapter: Background Material
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ABSTRACT: This chapter serves the rest of the book: all later chapters presuppose it. It introduces the calculus of binary relations, and relates it to basic concepts and results from lattice theory, universal algebra, category theory and logic. It also fixes the notation and terminology to be used in the rest of the book. Our aim here is to write in a way accessible to readers who desire a gentle introduction to the subject of relational methods. Other readers may prefer to go on to further chapters, only referring back to Chapt. 1 as needed. 
Article: Varieties of Lattices
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ABSTRACT: An interesting problem in universal algebra is the connection between the internal structure of an algebra and the identities which it satisfies. The study of varieties of algebras provides some insight into this problem. Here we are concerned mainly with lattice varieties, about which a wealth of information has been obtained in the last twenty years. We begin with some preliminary results from universal algebra and lattice theory. The second chapter presents some properties of the lattice of all lattice subvarieties. Here we also discuss the important notion of a splitting pair of varieties and give several characterizations of the associated splitting lattice. The more detailed study of lattice varieties splits naturally into the study of modular lattice varieties and nonmodular lattice varieties, dealt with in the third and fourth chapter respectively. Among the results discussed there are Freese's theorem that the variety of all modular lattices is not generated by its finite members, and several results concerning the question which varieties cover a given variety. The fifth chapter contains a proof of Baker's finite basis theorem and some results about the join of finitely based lattice varieties. Included in the final chapter is a characterization of the amalgamation classes of certain congruence distributive varieties and the result that  [Show abstract] [Hide abstract]
ABSTRACT: The poset product construction is used to derive embedding theorems for several classes of generalized basic logic algebras (GBLalgebras). In particular it is shown that every npotent GBLalgebra is embedded in a poset product of finite npotent MVchains, and every normal GBLalgebra is embedded in a poset product of totally ordered GMValgebras. Representable normal GBLalgebras have poset product embeddings where the poset is a root system. We also give a Conrad–Harvey–Hollandstyle embedding theorem for commutative GBLalgebras, where the poset factors are the real numbers extended with −∞. Finally, an explicit construction of a generic commutative GBLalgebra is given, and it is shown that every normal GBLalgebra embeds in the conucleus image of a GMValgebra. 
Conference Paper: Domain and Antidomain Semigroups
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ABSTRACT: We axiomatise and study operations for relational domain and antidomain on semigroups and monoids. We relate this approach with previous axiomatisations for semirings, partial transformation semi groups and dynamic predicate logic.  [Show abstract] [Hide abstract]
ABSTRACT: It is shown that the Boolean center of complemented elements in a bounded in tegral residuated lattice characterizes direct decompositions. Generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, we show that FLwalgebras decompose as a poset product over any finite set of join irreducible strongly central elements, and that bounded npotent GBLalgebras are represented as Esakia products of simple npotent MValgebras.  [Show abstract] [Hide abstract]
ABSTRACT: Generalized basic logic algebras (GBLalgebras for short) have been introduced in [JT02] as a generalization of Hájek’s BLalgebras, and constitute a bridge between algebraic logic and ℓgroups. In this paper we investigate normal GBLalgebras, that is, integral GBLalgebras in which every filter is normal. For these structures we prove an analogue of Blok and Ferreirim’s [BF00] ordinal sum decomposition theorem. This result allows us to derive many interesting consequences, such as the decidability of the universal theory of commutative GBLalgebras, the fact that npotent GBLalgebras are commutative, and a representation theorem for finite GBLalgebras as poset sums of GMValgebras, a result which generalizes Di Nola and Lettieri’s [DL03] representation of finite BLalgebras. 

Article: Topological duality and lattice expansions, I: A topological construction of canonical extensions
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ABSTRACT: The two main objectives of this paper are (a) to prove purely topological duality theorems for semilattices and bounded lattices, and (b) to show that the topological duality from (a) provides a construction of canonical extensions of bounded lattices. In previously known dualities for semilattices and bounded lattices, the dual spaces are compact 0dimensional spaces with additional algebraic structure. For example, semilattices are dual to 0dimensional compact semilattices. Here we establish dual categories in which the spaces are characterized purely in topological terms, with no additional algebraic structure. Thus the results can be seen as generalizing Stone's duality for distributive lattices rather than Priestley's. The paper is the first of two parts. The main objective of the sequel is to establish a characterization of lattice expansions, i.e., lattices with additional operations, in the topological setting built in this paper.  [Show abstract] [Hide abstract]
ABSTRACT: Petr Hajek identified the logic BL, that was later shown to be the logic of continuous tnorms on the unit interval, and defined the corresponding algebraic models, BLalgebras, in the context of residuated lattices. The defining characteristics of BLalgebras are representability and divisibility. In this short note we survey re cent developments in the study of divisible residuated lattices and attribute the inspiration for this investigation to Petr Hajek.
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